Computing roots of graphs is hard
Discrete Applied Mathematics
Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard
Theoretical Computer Science
Identifying and locating-dominating codes on chains and cycles
European Journal of Combinatorics
On a new class of codes for identifying vertices in graphs
IEEE Transactions on Information Theory
On the size of identifying codes in triangle-free graphs
Discrete Applied Mathematics
Minimum sizes of identifying codes in graphs differing by one vertex
Cryptography and Communications
Identifying path covers in graphs
Journal of Discrete Algorithms
Maximum size of a minimum watching system and the graphs achieving the bound
Discrete Applied Mathematics
Minimum sizes of identifying codes in graphs differing by one edge
Cryptography and Communications
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An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand, I. Charon, O. Hudry and A. Lobstein that if a graph on n vertices with at least one edge admits an identifying code, then a minimal identifying code has size at most n-1. They introduced classes of graphs whose smallest identifying code is of size n-1. Few conjectures were formulated to classify the class of all graphs whose minimum identifying code is of size n-1. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided.